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\section{A dynamic theory of the supply \& demand curves}\label{sec:dynamics}  \subsection{Definitions}  The classical supply and demand curves $S(p,t)$ and $D(p,t)$ (SD) represent respectively the amount of supply and demand that would reveal themselves   if the price were to be set to $p$ at time $t$. In classical Walrasian auctions, the equilibrium price $p_t^*$ is then set to the value that matches both quantities so that $D(p_t^*,t)=S(p_t^*,t)$. This equilibrium is unique provided the curves are strictly monotonous\footnote{By definition, or simply by common sense, the demand curve is a decreasing function of the price whereas the supply curve is increasing.}. The supply and demand curves, as well as the resulting equilibrium price, are represented on Figure \ref{fig:SD} (left).  In order to define the dynamics of the supply and demand curves, we also introduce the \emph{marginal supply and demand curves} (MSD), on which we will focus in the   rest of this paper. They are defined as the derivative of the SD curves   \be\nonumber  \begin{aligned}  &\rho_S(p,t)=\partial_p S(p,t)\geq 0;\\   &\rho_D(p,t)=-\partial_p D(p,t) \geq 0,  \end{aligned}  \ee  \noindent with the following interpretation: For any price $p$, $\rho_S(p,t)\text{d}p$ (resp. $\rho_D(p,t) \text{d}p$) is, at time $t$, the quantity of supply (resp. demand) that would materialize if the price changed from $p$ to $p+\text{d}p$ (resp. $p-\text{d}p$). The MSD curves can thus be seen as the density of supply and demand intentions in the vicinity of a given price. Figure \ref{fig:SD} (right) shows MSD curves corresponding to the SD curves: Higher MSD levels correspond to larger slopes for the SD curves.      In the Walrasian story, supply and demand pre-exist and the Walrasian auctioneer gropes ({\it t\^atonne}) to find the price $p_t^*$ that maximizes the amount of possible transactions. The auction then takes place at time $t$ and removes instantly all matched orders. Assuming that all the supply and demand intentions close   to the transaction price were revealed before the auction and were matched, the state of the MSD just after the auction is simple to describe, see Figures \ref{fig:SD} \& \ref{fig:scheme}:  \be\label{eq:after}  \left\lbrace   \begin{array}{ccc}  \rho_S(p,t^+) &= &\rho_S(p,t^-) \quad (p > p_t^*)\\  & = &0 \quad (p \leq p_t^*)\\  \rho_D(p,t^+) &= &\rho_D(p,t^-) \quad (p < p_t^*)\\  &=&0 \quad (p \geq p_t^*).\\  \end{array}  \right.  \ee  But what happens next, once the auction has been settled? So far the story does not tell (to the best of our knowledge). The aim of the following is to set up a   general framework for the dynamics of the supply and demand curves. This will allow us to describe, among other questions,   how the supply and demand curves evolve from the truncated shape given by Equation (\ref{eq:after}) up to the next auction   at time $t + \tau$ (where $\tau$ is the inter-auction time).  \subsection{General hypotheses about the behaviour of agents}\label{sec:behaviour}  The theory that we present here relies on weak and general assumptions on agents behaviours that translate into a simple and universal   evolution of the MSD curves, with only very few parameters\footnote{In fact, as shown in \cite{donier2014fully} and below, only two parameters suffice to describe the problem in the vicinity of the price:   one is the price volatility, and the other one is related to market activity (traded volume per unit time).}. The MSD curves aggregate the intentions of all agents, which would materialize in the   ``real'' order book if it was not for fear of being picked off by more informed traders, or of revealing some information to the market. This is why the MSD  curves were called the ``latent'' order book (LOB) in Refs. \cite{Iacopo:2013,Iacopo:2014,donier2014fully}, as initially proposed in \cite{Toth:2011}.   We will assume that there is a so-called ``fundamental'' price process $\widehat p_t$ which is only partially known to agents, in a sense clarified below (see Section \ref{sec:discussion}). For simplicity, we will also posit that $\widehat p_t$ is an additive Brownian motion. In the absence of transactions, the MSD curves evolve according to three distinct mechanisms, that we model as follows:  \begin{itemize}  \item New intentions, not present in the supply and demand before time $t$, can appear. The probability for new buy/sell intentions to appear between $t$ and $t + {\rm d}t$ and between prices $p$ and $p+{\rm d}p$   is chosen to be $\omega_\pm(p - \widehat p_t)$, where $\omega_+(x)$ is   a decreasing function of $x$ and $\omega_-(x)$ is an increasing function of $x$.  \item Already existing intentions to buy/sell at price $p$ can simply be cancelled and disappear from the supply and demand curves. The probability for an existing buy/sell intention   around price $p$ to disappear between $t$ and $t + {\rm d}t$ is chosen to be $\nu_\pm(p - \widehat p_t)$.  \item Already existing intentions to buy/sell at price $p$ can be revised. Between $t$ and $t + {\rm d} t$, each agent $i$ revises his/her reservation price $p^i$ to $p^i + \beta^i {\rm d}\xi_t + {\rm d}W_{i,t}$, where ${\rm d} \xi_t$   is common to all $i$, representing some public information. $\beta^i$ is the sensitivity of agent $i$ to the news, which we imagine to be a random variable   from agent to agent, with a mean normalized to $1$. Some agents may over-react ($\beta^i > 1$), others under-react ($\beta^i < 1$).   The idiosyncratic contribution ${\rm d} W_{i,t}$ is an independent Wiener noise both across different agents and in time,   with distribution of mean zero\footnote{One could generalize the calculations below to the case where the mean is non zero (modelling for example the tendency  of agents to revise their reservation price in the direction of the traded price). This would affect none of the conclusions below, at least in the limit where the   inter-auction time $\tau$ becomes very small.} and variance $\Sigma_i^2 {\rm d} t$, that may depend on the agent (some agents might be more ``noisy'' than others).   \end{itemize}  We will furthermore assume that the ``news'' term ${\rm d}\xi_t$ is a Wiener noise of variance $\sigma^2 \d t$, corresponding to a Brownian motion for the fundamental price   $\widehat p_t = \int^t {\rm d}\xi_{t'}$ with volatility $\sigma$.   Normalising the mean of the $\beta^i$'s to unity thus corresponds to the assumption that agents are on average unbiased in their interpretation of the news -- i.e. their intentions remain  centred around the fundamental price $\widehat p_t$ in the course of time -- but see the expanded discussion of this point in Section \ref{sec:discussion}.   Our central assumptions are \emph{heterogeneity}, together with the hypothesis that idiosyncratic behaviours ``average out'' in the limit of a very large number of participants, i.e., no single agent accounts for a finite fraction   of the total supply or demand. While not strictly necessary, this assumption leads to a deterministic aggregate behaviour and allows one to gloss over some rather involved mathematics.  \subsection{The model in terms of optimizing agents}  \label{sec:agent-opt}  The above assumptions might appear obscure to those used to think in terms of rational optimizing agents and equilibria. Here we rephrase these assumptions in a language closer to standard economic intuition.  We consider an \emph{open} economic system, in which many heterogeneous, infinitesimal agents operate. Each agent $i$ has a certain utility   $\mathcal{U}_i(p,\theta|\widehat{p}_t^i,{\cal F}_t)$ for buying ($\theta=+1$) or selling ($\theta=-1$) a unit (small) quantity   at price $p$, given his/her estimate of the fundamental price $\widehat{p}_t^i$ and all the information about the rest of the world, available at time $t$, encoded in   ${\cal F}_t$. The third option available to agent $i$ is to be inactive ($\theta=0$), in which case the number of goods s/he owns remains constant.   Agents are heterogeneous in the sense that both their utility function and their estimates of the fundamental price are different; one can think of them as random members   of some adequate statistical ensembles. For the sake of simplicity, we consider no interest rate and no risk of any kind.  At time $t$, each agent computes his optimal action $p_t^i, \theta_t^i$ as the result of the following optimisation program:  \be\label{opt}  (p_t^i,\theta_t^i) = \underset{p,\theta}{\text{argmin}}~~\mathcal{U}_i(p,\theta|\widehat{p}_t^i,{\cal F}_t).  \ee  Because of the random evolution of the outside world summarized by $\widehat{p}_t^i,{\cal F}_t$, the value of $\theta_t^i \in \{-1,0,+1\}$ can change between  $t$ and $t + \d t$. For the sake of simplicity, we assume that the change of the state of the world in time $\d t$ is never so large as to induce direct   transitions from $\mp 1 \to \pm 1$ without pausing at $0$. Hence, between $t$ and $t + \d t$, the following transitions (or absence thereof) are possible:  \begin{itemize}  \item $0 \to 0$: this clearly induces no change in the MSD curves;  \item $0 \to \pm 1$: in this case, agent $i$ previously absent from the market becomes either a buyer or a seller, with reservation price $p_t^i$ given by  Equation (\ref{opt}). The assumption that agents are heterogeneous translates in a model where this event is a Poisson process with some arrival rate $\omega_\pm(p)$;  \item $\pm 1 \to 0$: in this case, agent $i$ previously present in the market as a buyer or a seller, decides to become neutral, which is modelled as a Poisson process with some cancellation rate $\nu_\pm(p)$;  \item $\pm 1 \to \pm 1$: in this case, a buyer/seller remains a buyer/seller, but may change his/her reservation price because the solution of Equation (\ref{opt})  has changed. Writing $p_t^i = f_i(\widehat{p}_t^i,t)$, where $f$ is a regular function if $\mathcal{U}_i$ is regular enough, and applying It\^{o}'s lemma, one finds:  \be\nonumber  \begin{aligned}  \d p_t^i &= \frac{\partial f_i}{\partial t} \d t + \frac{\partial f_i}{\partial p} \d \widehat{p}_t^i + \frac{\sigma_i^2}{2}  \frac{\partial^2 f_i}{\partial p^2} \d t \\  &= \alpha_t^i \d \widehat{p}_t^i + \gamma_t^i \d t.  \end{aligned}  \ee  The drift term $\gamma_t^i$ will play little role in the following (see previous footnote), and we neglect it henceforth. In order to recover the specification  of the above section, we further decompose the price revision $\d p_t^i = \alpha_t^i \d \widehat{p}_t^i$ into a \emph{common} component $\beta^i {\rm d}\xi_t$   and an \emph{idiosyncratic} component ${\rm d}W_{i,t}$ as above.  \end{itemize}  Therefore, the mechanism proposed in the above section indeed describes the behaviour of an open system of \emph{infinitesimal} and \emph{heterogeneous} market  participants. Note that we do not need to distinguish between fundamental investors, noise traders and market makers, as for example in the Kyle model \cite{kyle1985continuous}. This is due to our assumption that the market contains a very large number of participants, in which case the MSD curves are   continuous. Discretization effects (in price and in quantity) would open gaps in the MSD curves, and specific market makers would then be needed to ensure continuous, orderly trading. Finally, and quite importantly, the price dynamics in the above setting is arbitrage free (see \cite{donier2014fully}). There is therefore   no optimal strategic component that is missing from the above utility maximisation program.  \subsection{The ``free evolution'' equation for the MSD curves}\label{sec:free}  Endowed with the above hypothesis, one can derive stochastic partial differential equations for the evolution of the marginal supply ($\rho_S(p,t)=\partial_p S(p,t)$) and the marginal   demand ($\rho_D(p,t)=-\partial_p D(p,t)$) in the absence of transactions \cite{donier2014fully}.  It turns out that, as expected, these equations take a simpler form in the reference frame of the (moving) fundamental price $\widehat p_t$. Introducing the shifted price $y = p - \widehat p_t$,   one finds \cite{donier2014fully}:\footnote{See also \cite{lasry2007mean, lehalle2011high} for similar ideas in the context of mean-field games.   Note that Equation (\ref{eq:dynamics}) is strictly valid when $\rho_S(p,t)$ and $\rho_D(p,t)$ are be interpreted as the marginal supply and demand curves averaged over the noise processes.   Otherwise some noisy component remains, see e.g. \cite{dean1996langevin}.}  \be\label{eq:dynamics}  \left\lbrace   \begin{array}{ccc}  \partial_t \rho_D(y,t) &=& {\cal D} \partial^2_{yy} \rho_D(y,t) - \nu_+(y) \rho_D(y,t) + \omega_+(y); \\  \partial_t \rho_S(y,t) &=& \underbrace{\D \partial^2_{yy} \rho_S(y,t)}_{\text{Updates}} -   \underbrace{\nu_-(y) \rho_S(y,t)}_{\text{Cancellations}} + \underbrace{\omega_-(y)}_{\text{New orders}},\\  \end{array}  \right.  \ee  where ${\cal D} = \frac12 [\E_i(\Sigma_i^2) + \sigma^2 {\text{Var}}(\beta^i)]$, i.e. part of the diffusion term comes from the purely idiosyncratic ``noisy" updates of agents ($\E_i(\Sigma_i^2)$), and another part comes from the   inhomogeneity of their reaction to news ($\sigma^2 {\text{Var}}(\beta^i))$, which indeed vanishes if all $\beta^i$'s are equal to unity.\footnote{Here we neglect the possibility that buyers and sellers update their  price differently, but one could make a distinction between a ${\cal D}_+$ and a ${\cal D}_-$, or even make ${\cal D}$ price/time dependent.}   These equations, that are at the core of the present paper, describe the structural   evolution of supply and demand around the fundamental price $\widehat{p}_t$. Notice however that $\widehat{p}_t$ has disappeared from the above equations. The dynamics of the MSD curves can be treated independently from the dynamics of the price itself, provided one describes the MSD in the reference frame of the price. There is however a direct relationship between the price volatility $\sigma$ and the diffusion coefficient ${\cal D}$, as expressed above and noted in \cite{donier2014fully}.  Interestingly, whereas the price is random and follows a rough path (typically a Brownian motion), the structural part is deterministic and smooth, thanks to the assumption of ``infinitesimal'' orders (that can be made rigourous by considering an appropriate scaling for system parameters that corresponds to a hydrodynamic limit, see \cite{gao2014hydrodynamic}).   The above equations for $\rho_D(y,t)$ and $\rho_S(y,t)$ are linear and can be formally solved in the general case, starting from an arbitrary initial condition such as Equation (\ref{eq:after}), using a spectral decomposition of  the evolution operator. This general solution is however not very illuminating, and we rather focus here in the special case where $\nu_\pm(y) \equiv \nu$ does not depend on $y$ nor on the side of the latent order book. The general solution can then be written in a fairly transparent way, as:  \be \label{eq:gen-sol}  \rho_{S,D}(y,t) = \int_{-\infty}^{+\infty} \frac{\d y'}{\sqrt{4 \pi \D t}} \, \rho_{S,D}(y',t=0^+) e^{-\frac{(y'-y)^2}{4 \D t} - \nu t}   + \int_0^t \d t' \int_{-\infty}^{+\infty} \frac{\d y'}{\sqrt{4 \pi \D (t-t')}} \, \omega_{\pm}(y') e^{-\frac{(y'-y)^2}{4 \D (t-t')} - \nu (t-t')},  \ee  where $\rho_{S,D}(y,t=0^+)$ is the initial condition, i.e. just after the last auction.   We will now explore the properties of the above solution at time $t= \tau^-$, i.e., just before the next auction, in the two asymptotic limits $\tau \to \infty$, corresponding to very infrequent auctions, and $\tau \to 0$, corresponding to continuous time auctions.  \section{Discrete Auctions and Price Impact}\label{sec:theory}  The aim of this section is to show that the shape of the marginal supply and demand curves can be fully characterized in the limit of very infrequent auctions (corresponding to Walras' auctions) and in the opposite limit of nearly continuous time  auctions (corresponding to financial markets), and describe the transition between the two limits. The upshot is that while the liquidity around the auction price is in general finite and leads   to a linear impact using the standard argument in Equation (\ref{Kyle-lambda1}) above, this liquidity vanishes as $\sqrt{\tau}$ when the inter-auction time $\tau \to 0$. This signals the breakdown of linear impact and, as shown at the end of the section,   its replacement by the square-root law mentioned in the introduction.   \subsection{Walras, or the limit of infrequent auctions}  Letting $t=\tau \to \infty$ in the above Equation (\ref{eq:gen-sol}), one immediately sees that the first term disappears, meaning that one reaches a {\it stationary solution} $\rho^{\text{st.}}_{S,D}(y)$ that is independent  of the initial condition. The second term can be simplified further to give the following general solution:  \be\label{eq:st-sol}  \rho^{\text{st.}}_{S,D}(y) = \frac{1}{2 \sqrt{\nu \D}} \int_{-\infty}^{+\infty} \d y' \omega_{\pm}(y') \, e^{-\sqrt{\frac{\nu}{\D}}|y'-y|}.  \ee  A particularly simple case is when $\omega_{\pm}(y)=\Omega_\pm e^{\mp \mu y}$, meaning that buyers(/sellers) have an exponentially small probability to be interested in a transaction at high/low prices. In this toy-example, one  readily finds that a stationary state only exists when $\nu > \D \mu^2$ and reads:  \be\nonumber  \rho^{\text{st.}}_{S,D}(y) = \frac{\Omega_\pm}{\nu - \D \mu^2} e^{\mp \mu y}.  \ee  Other forms for $\omega_{\pm}(y)$ can be investigated as well, for example $\omega_{\pm}(y) = \omega_{\pm}^0\mathbbm{1}_{\{y\overset{>}{\underset{<}{ }}0\}}$ which yields:  \be\nonumber  \rho^{\text{st.}}_{S,D}(y) = \frac{\omega_{\pm}^0}{2\nu}\left[ 1\pm\text{sign}(y)(1-e^{-\sqrt{\nu/D}\mid y \mid}) \right],  \ee  that we will use in Figures \ref{fig:books} and \ref{fig:vanishing}. The shape of $\rho^{\text{st.}}_{S,D}(y)$ is generically the one shown in Figure \ref{fig:SD} with an overlapping region where buy/sell orders coexist.   The auction price $p^*_\tau = \widehat p_\tau + y^*$ is determined by the condition $D(p_\tau^*,\tau^-)=S(p_\tau^*,\tau^-)$, or else:  \be\nonumber  \int_{y^*}^\infty \d y \rho^{\text{st.}}_{D}(y) = \int_{-\infty}^{y^*} \d y \rho^{\text{st.}}_{S}(y) \equiv v^*,  \ee  where $v^*$ is the volume exchanged during the auction. For the simple exponential case above, this equation can be readily solved as:   \be \label{eq:lin-imp}\nonumber  y^* = \frac{1}{2\mu} \ln \frac{\Omega_+}{\Omega_-},  \ee  with a clear interpretation: if the new buy order intentions accumulated since the last auction happen to outsize the new sell intentions during the same period, the auction price will exceed the fundamental price,   and vice-versa. This pricing error is expected to be small if the order book is observable during the inter-auction period, since in that case $\Omega_+$ and $\Omega_-$ will track each other and remain close.   Otherwise, one expects the imbalance to invert in the next period, leading to a kind of ``bid-ask bounce'' well known in the context of market microstructure. One can also compute the volume exchanged   during the auction $v^*$. One finds:   \be\nonumber  v^* = \frac{\sqrt{\Omega_+ \Omega_-}}{\mu(\nu - \D \mu^2)}.  \ee  Just after the auction, the MSD curves start again   from $\rho^{\text{st.}}_{S,D}(y)$, truncated below (resp. above) $y^*$, as in Equation (\ref{eq:after}).  Let us now turn to price impact in this model. From Equation (\ref{eq:st-sol}), it is immediate that for any clearing price $y^*$, both $\rho^{\text{st.}}_{S}(y^*)$ and $\rho^{\text{st.}}_{D}(y^*)$ are strictly positive.   This would remain true even if the dependence on $y$ of cancellation rate $\nu_\pm(y)$ was reinstalled.   The general argument given in the introduction therefore predicts a {\it linear impact} for an extra buy/sell quantity given by:  \be\label{Kyle-lambda2}\nonumber  {\cal I}(Q) = \pm \lambda Q; \qquad \qquad \lambda =\frac{1}{\rho^{\text{st.}}_{S}(y^*)+\rho^{\text{st.}}_{D}(y^*)}.  \ee  For the exponential case, this again takes a simple form:  \be\nonumber  \lambda = \frac{{ \nu-\D \mu^2}}{2 \sqrt{\Omega_+ \Omega_-}},  \ee  whereas for a general symmetric order flow $\omega_{+}(y)=\omega_{-}(-y)$, $y^*$ is obviously equal to zero, leading to:  \be\nonumber  \lambda = \frac{\sqrt{\nu \D}}{\int_{-\infty}^{+\infty} \d y' \omega(y') e^{-\sqrt{\frac{\nu}{\D}}|y'|}}.  \ee  For $\omega_{\pm}(y) = \omega^0\mathbbm{1}_{\{y\overset{>}{\underset{<}{ }}0\}}$, one obtains the simple and intuitive result:   \be\nonumber  \lambda = \frac{\nu}{\omega_0},  \ee  i.e. that the market liquidity, measured by $\lambda^{-1}$, grows linearly with the rate of incoming orders and inversely proportionally to the   cancellation rate.  The main point of the present section is that when the inter-auction time is large enough, each auction clears an equilibrium supply with an equilibrium demand, with very simple and predictable outcomes. This corresponds to the quasi-static dynamics discussed in item 2., Section ~\ref{sec:eco}, and to the standard representation of market dynamics in the Walrasian context, since in this case only the long-term properties of supply and demand matter and the whole transients are discarded. The next section will depart from this limiting case, by introducing a finite inter-auction time such that the transient dynamics of supply and demand becomes a central feature in the theory.  \subsection{High frequency auctions}  We will now investigate the alternative limit where the inter-auction time $\tau$ tends to zero. Since all the supply (resp. demand) curve left (resp. right)   of the auction price is wiped out by the auction process, one expects intuitively that after a very small time $\tau$, the density of buy/sell orders in the  immediate vicinity of the transaction price will remain small. We will show that this is indeed the case, and specify exactly the shape of  the stationary MSD after many auctions have taken place. Consider again Equation (\ref{eq:gen-sol}) just before the $n+1$th auction at time $(n+1) \tau^-$, in the  case where the flow of new orders is symmetric, i.e. $\omega_{+}(y)=\omega_{-}(y)$, such that the transaction price is always at the fundamental price ($y^* = 0$). We will focus on  the supply side and postulate that $\rho_S(y,t=n \tau^-)$ can be written, in the vicinity of $y=0$, as \footnote{This approximation happens to be exact in the particular setting considered in \cite{donier2014fully}.}:  \be\label{eq:iteration}  \rho_S(y,t=n \tau^-) = \sqrt{\tau} \, \phi_n\left(\frac{y}{\sqrt{\D \tau}}\right)+ O(\tau)  \ee  when $\tau \to 0$ (and symmetrically for the demand side). Plugging this ansatz into Equation (\ref{eq:gen-sol}), making the change of variable $y' \to \sqrt{\D \tau} w$   and taking the limit $\tau \to 0$ leads to the following iteration equation, exact up to order $\sqrt{\tau}$:\footnote{An extra correction of order $\sqrt{\tau}$ would appear if a drift term was added to Equation (\ref{eq:dynamics}).}  \be\nonumber  \phi_{n+1}(u) = \int_0^{+\infty} \frac{\d w}{\sqrt{4 \pi}} \phi_n(w) e^{-(u-w)^2/4} + \sqrt{\tau} \, \omega(0) + O(\tau).  \ee  Note that $\nu$ has entirely disappeared from the equation (but will appear in the boundary condition, see below), and only the value of $\omega$ close to the transaction price is relevant at this order.  After a very large number of auctions, one therefore finds that the stationary shape of the demand curve close to the price and in the limit $\tau \to 0$ is given by the non-trivial solution of the following   fixed point equation:  \be\label{eq:atkinson}  \phi_\infty(u) = \int_0^{+\infty} \frac{\d w}{\sqrt{4 \pi}} \phi_\infty(w) e^{-(u-w)^2/4},  \ee  supplemented by the boundary condition $\phi_\infty(u \gg 1) \approx \L \sqrt{\D} u$, where $\L$ is a constant to be determined below. [Note that the solution of Equation (\ref{eq:atkinson}) is determined up to a   multiplicative factor that must be fixed by some external condition].    \begin{figure}[t!]  \begin{center}  \includegraphics[height=7.5cm]{images/atkinson}  \end{center}  \caption{Graph of the normalized exact solution $\phi_\infty(u)$, and its affine approximation. The whole picture must be rescaled by a factor $\sqrt{\tau}$ to recover the order book when the inter-auction time is $\tau$. The hatched region corresponds to the volume to be executed, and therefore scales with $\tau$.}  \label{fig:atkinson}  \end{figure}  \begin{figure}[t!]  \begin{center}  \includegraphics[height=7.5cm]{images/books}  \end{center}  \caption{Left: Shape of the marginal supply curve immediately before the auctions, for different inter-auction times $\tau$, in the case $\omega_{\pm}(y) = \omega_{\pm}^0\mathbbm{1}_{\{y\overset{>}{\underset{<}{ }}0\}}$. Right: Shape of the MSD immediately after the auctions, again for different   inter-auction times $\tau$. Note that as $\tau \to 0$, the MSD acquires a characteristic V-shape.}  \label{fig:books}  \end{figure}  Equation (\ref{eq:atkinson}) is of the Wiener-Hopf type and its analytical solution can be found in \cite{atkinson1974wiener,boersma1974note}. We plot numerically this solution in Figure \ref{fig:atkinson}; it is seen to be   numerically very close to an affine function for $u > 0$: $\phi_\infty(u) \approx \L \sqrt{\D} (u + u_0)$ with $u_0 \approx 0.824$. In summary, the stationary shape $\rho_{S,\text{st.}}(y)$ of the marginal supply curve in the frequent auction limit $\tau \to 0$ and close to the transaction price ($y = O(\sqrt{\D \tau})$,   has a {\it universal shape}, independent of the detailed specification of the model (i.e., the functions $\nu_\pm(y)$ and $\omega_\pm(y)$). This supply curve is given by   $\sqrt{\tau} \phi_\infty(y/\sqrt{\D \tau})$, which can itself be approximated by a simple affine function that will fully suffice for the purpose of the present paper:  \be\label{eq:final}  \rho_{S,\text{st.}}(y \geq 0) \approx \L (y + y_0); \qquad y_0 = u_0 \sqrt{\D \tau}; \qquad (\tau \to 0),  \ee  and similarly for $\rho_{D,\text{st.}}(y)$, see Figure \ref{fig:books}. The detailed interpretation of this result -- in terms of market liquidity and price impact -- will be given below.   We however still need to find the value of $\L$. This is done by comparing with the stationary solution $\varphi_{\text{st.}}(y)$ of Equation (\ref{eq:dynamics}) that satisfies the boundary solution   $\varphi_{\text{st.}}(0)=0$ (valid for $\tau=0$). For $\nu_\pm(y) = \nu$, $\varphi_{\text{st.}}(y)$ can be computed explicitly and is given by:  \be\nonumber  \varphi_{\text{st.}}(y) = \frac{1}{\D} e^{-\sqrt{\nu/\D}\, y} \int_0^y {\rm d}y' e^{2\sqrt{\nu/\D} \,y'} \int_{y'}^\infty {\rm d}y'' e^{-\sqrt{\nu/\D} \,y''}\omega(y'').  \ee  Expanding $\varphi_{\text{st.}}(y)$ for small $y$ (but still much larger than $\sqrt{\D \tau}$) finally leads to:  \be\nonumber  \varphi_{\text{st.}}(y) \approx \L y; \qquad \L = \frac{1}{\D} \int_{0}^\infty {\rm d}y' e^{-\sqrt{\nu/\D} \,y'}\omega(y'),  \ee  where $\L$ can be seen as a measure of the market liquidity (see \cite{donier2014fully} and below). Again in the simple case $\omega_{\pm}(y) = \omega^0\mathbbm{1}_{\{y\overset{>}{\underset{<}{ }}0\}}$, one  finds:  \be\nonumber  \L = \frac{\omega_0}{\sqrt{\nu {\cal D}}}.  \ee  Therefore, liquidity increases with the order arrival rate and decreases with their cancellation rate, as above, but also decreases   with the diffusion constant ${\cal D}$ that can be loosely identified with market volatility (see the discussion above).